$L$-function for $\mathrm{Sp}(4)\times \mathrm{GL}(2)$ via a non-unique model

Abstract

In this paper we prove a conjecture of Ginzburg and Soudry on an integral representation for the $L$-function $L^S(s,\ddot{I}€\times \ddot{I}„)$ attached to a pair $(\ddot{I}€,\ddot{I}„)$ of irreducible automorphic cuspidal representations of ${\mathrm{Sp}}_4(\mathbb{A})$ and ${\mathrm{GL}}_2(\mathbb{A})$, which is derived from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis. Two applications are given. First, we relate the existence of the poles of $L^S(s,\ddot{I}€\times \ddot{I}„)$ to the non-vanishing of certain period integrals. Second, for certain family of cuspidal representations, we prove that $L^S(s,\ddot{I}€\times \ddot{I}„)$ is holomorphic.

37 pages